I am trying to read a book on Linear Delay Differential Equation, but I am getting stuck with some passages in some important proofs. I tried looking for alternative versions in Internet but couldn't find anything useful. Any help (or reference) is appreciated.
Let $\{T(t)\}_{t\in\mathbb{R}^+}$ be a strongly continuous semigroup of bounded linear operators, and let $A$ be its generator. Let us assume that $A$ is closed or bounded.
Let $\lambda\in\Bbb{C}$ be an eigenvalue for $A$, and let $\varphi,\psi:[-\tau,0]\to$ be such that $$(A-\lambda I)\varphi=\psi$$ and $$A\varphi=\varphi'$$ (that is, the semigroup $\{T(t)\}_{t\in\mathbb{R}^+}$ is the semigroup of the solution operators). The book says that from this two conditions we can derive that $$\varphi(\theta)=e^{\lambda\theta}\varphi(0)+\int_0^{\theta}e^{\lambda(\theta-s)}\psi(s) ds$$ for any $\theta\in[\tau,0]$. I really can't see why this should be true.